Problem 75 - Intermediate Algebra

Question

Assume that there are no deposits or withdrawals. Comparison of Compounding Methods. An initial deposit of \(\$ 5,000\) grows at an annual rate of \(8.5 \%\) for 5 years. Compare the final balances resulting from annual compounding and continuous compounding.

Step-by-Step Solution

Answer

The final balance is \( \$7517.10 \) with annual compounding and \( \$7647.55 \) with continuous compounding.

Step 1: Understanding Annual Compounding

In annual compounding, interest is added to the principal at the end of each year. The formula to calculate the future value with annual compounding is:\[ FV = P \times (1 + r)^t \]where \( P = 5000 \) is the initial deposit, \( r = 0.085 \) is the annual interest rate, and \( t = 5 \) is the number of years.

Step 2: Calculating Annual Compounding Result

Using the formula for annual compounding:\[ FV = 5000 \times (1 + 0.085)^5 \]Calculate:\[ FV = 5000 \times (1.085)^5 \]\[ FV = 5000 \times 1.50342 \]\[ FV = 7517.10 \]Thus, the final balance with annual compounding is \( \$7517.10 \).

Step 3: Understanding Continuous Compounding

In continuous compounding, interest is added continuously to the principal. The formula to calculate future value with continuous compounding is:\[ FV = P \times e^{rt} \]where \( e \approx 2.71828 \), \( P = 5000 \), \( r = 0.085 \), and \( t = 5 \).

Step 4: Calculating Continuous Compounding Result

Using the formula for continuous compounding:\[ FV = 5000 \times e^{(0.085 \times 5)} \]Calculate the exponent:\[ FV = 5000 \times e^{0.425} \]\[ FV = 5000 \times 1.52951 \] (using \( e^{0.425} \approx 1.52951\))\[ FV = 7647.55 \]Thus, the final balance with continuous compounding is \( \$7647.55 \).

Step 5: Comparing the Results

With annual compounding, the balance is \( \\(7517.10 \) and with continuous compounding, it’s \( \\)7647.55 \). The continuous compounding yields a slightly higher amount.

Key Concepts

Annual Compounding

When an investment or savings grow through annual compounding, interest is calculated and added to the principal balance once every year. This process results in exponential growth of the investment value, because each year, the interest earns its own interest. Essentially, you're earning interest on the interest that has been previously added.
This concept can be captured by the formula:

• \[ FV = P \times (1 + r)^t \]

where

• \( P \) is the initial principal balance,
• \( r \) is the annual interest rate,
• \( t \) is the time the money is invested for.

In our example, starting with a principal of \(5,000 at an interest rate of 8.5% over 5 years, the value would grow to approximately \( \\)7,517.10 \) through annual compounding. Every year, your balance grows not just from the original deposit but from all previous interest accrued as well.

Continuous Compounding

With continuous compounding, interest isn't just applied once at the end of the year or any set period. Instead, it's added at every possible moment. This means that your principal grows at every conceivable smaller fraction of time, leading to a higher final amount compared to traditional compounding methods.
The formula used for continuous compounding involves the mathematical constant \( e \), roughly equal to 2.71828:

• \[ FV = P \times e^{rt} \]

Where

• \( e \) represents the base of the natural logarithm,
• \( P \), \( r \), \( t \) have the same meanings as in annual compounding.

In the exercise, applying an 8.5% interest rate continuously to \(5,000 over 5 years, the balance grows to approximately \( \\)7,647.55 \). The effect of compounding interest continuously allows even more growth beyond what annual compounding provides.

Exponential Growth

Both annual and continuous compounding illustrate the principle of exponential growth in finance. Exponential growth refers to the process in which the rate of growth of a quantity is proportional to its current value. In the context of investment, the larger the principal and interest accrued, the greater the growth over time.
This growth pattern is reflected in the compounding formulas:

• Annual Compounding: \[ FV = P \times (1 + r)^t \]
• Continuous Compounding: \[ FV = P \times e^{rt} \]

These formulas show how the future value (FV) of money can increase rapidly as time and interest compound. The key driver is the effect of earning interest on both the initial investment and past accumulations.
In our example, both compounding methods result in a substantial increase in the principal amount. Quite simply, exponential growth through compounding is a powerful force, underscoring why starting to invest or save early can be particularly beneficial. Continuous compounding generally offers slightly more growth than the annual method because it frequently adds interest to the principal.